Near-field Acoustical Holography (NAH) is a very useful tool for 3D visualization of sound radiation and for precise noise source localization based on measurements over a surface near the sound source. Its ability to reconstruct also the evanescent wave components ensures a very high spatial resolution.
A known Near-field Acoustical Holography method is based on regular-grid measurements across a level surface in a separable coordinate system, allowing the NAH calculation to be performed by spatial Discrete Fourier Transform (DFT), see e.g. E. G. Williams, J. D. Maynard, and E. J. Skudrzyk, “Sound source reconstruction using a microphone array,” J. Acoust. Soc. Am. 68, 340-344 (1980). Due to the use of DFT, the processing is very fast, but a side effect of using the DFT includes severe spatial windowing effects unless the measurement area fully covers the areas with high sound pressure. In some cases this requirement on the measurement area cannot be fulfilled, and in many cases the necessary size becomes prohibitively large.
A set of techniques have been proposed to reduce the spatial windowing effects, while still maintaining the DFT spatial processing but at the cost of an increased complexity and computational demands, see e.g. J. Hald, “Reduction of spatial windowing effects in acoustical holography,” Proceedings of Inter-Noise 1994. Typically an iterative procedure is first used to extrapolate the measured sound pressure outside the measured area, followed by application of a DFT based holography method on the extended data window.
Other methods have been proposed that seek to avoid the use of spatial DFT and to provide a reduction in the required measurement area.
One such method is the Helmholtz' Equation Least Squares (HELS) method which uses a local model of the sound field in terms of spherical wave functions, see e.g. U.S. Pat. No. 6,615,143, Z. Wang and S. F. Wu, “Helmholtz equation-least-squares method for reconstructing the acoustic pressure field,” J. Acoust. Soc. Am, 102(4), 2020-2032 (1997); or S. F. Wu, “On reconstruction of acoustic fields using the Helmholtz equation-least-squares method,” J. Acoust. Soc. Am, 107, 2511-2522 (2000). However, since only spherical wave functions with a common origin are used to represent the sound field, errors will be introduced in the sound field reconstruction on the source surface, unless the source surface is also spherical and centred in the same origin. Another drawback of this prior art method is the requirement of the estimation problem not to be under-determined, meaning that the number of measurement positions must be larger than or equal to the number of spherical wave functions used in the field representation. This can require a large number of measurement positions to obtain a sufficiently accurate model. A third drawback is that scaling of the wave functions is not applied in such a way that functions with stronger decay in the model region are scaled to lower amplitudes in the same region. Because of this lack of scaling, traditional regularization methods like Tikhonov regularization do not work properly. Instead, the above prior art method applies a computationally expensive iterative search for an optimal truncation of the spherical wave expansion combined with a least squares solution without regularization.
Another previously proposed methodology is the Statistically Optimized Near-field Acoustic Holography (SONAH) method suggested in R. Steiner and J. Hald, “Near-field Acoustical Holography without the errors and limitations caused by the use of spatial DFT,” Intern. J. Acoust. Vib. 6, 83-89 (2001). This prior art method is based on plane wave functions and the acoustic quantities on a mapping surface near the measurement surface are calculated by using a transfer matrix defined in such a way that all propagating waves and a weighted set of evanescent waves are projected with optimal average accuracy. The transfer matrix is obtained from 1) an Auto-correlation matrix of Auto- and Cross-correlations between all pairs of measurement positions and 2) a Cross-correlation matrix of Cross-correlations from each measurement point to every reconstruction position. The Auto- and Cross-correlations are in a domain of plane propagating and evanescent waves.
An application of this framework to specific planar measurement geometries has been described in J. Hald, “Patch near-field acoustical holography using a new statistically optimal method”, Proceedings of Inter-Noise 2003, and J. Hald, “Patch holography in cabin environments using a two-layer handheld array and an extended SONAH algorithm,” Proceedings of Euronoise 2006.
These prior art documents disclose the computation of the pressure and the particle velocity component normal to the measurement plane. The method is, however, computationally expensive, because numerical evaluation of the formulae for the correlations disclosed in these prior art documents is computationally inefficient.